theorem Th75:
  a,b // c,d implies (a,c // b,d or ex x st LIN a,c,x & LIN b,d,x )
proof
  assume
A1: a,b // c,d;
  assume
A2: not a,c // b,d;
A3: now
    consider z such that
A4: a,b // c,z and
A5: a,c // b,z by DIRAF:40;
    assume
A6: a<>b;
A7: now
      c,d // c,z by A1,A6,A4,AFF_1:5;
      then LIN c,d,z by AFF_1:def 1;
      then LIN d,c,z by AFF_1:6;
      then d,c // d,z by AFF_1:def 1;
      then z,d // d,c by AFF_1:4;
      then consider t such that
A8:   b,d // d,t and
A9:   b,z // c,t by A2,A5,DIRAF:40;
      assume b<>z;
      then a,c // c,t by A5,A9,AFF_1:5;
      then c,a // c,t by AFF_1:4;
      then LIN c,a,t by AFF_1:def 1;
      then
A10:  LIN a,c,t by AFF_1:6;
      d,b // d,t by A8,AFF_1:4;
      then LIN d,b,t by AFF_1:def 1;
      then LIN b,d,t by AFF_1:6;
      hence thesis by A10;
    end;
    now
      assume b=z;
      then b,a // b,c by A4,AFF_1:4;
      then LIN b,a,c by AFF_1:def 1;
      then
A11:  LIN a,c,b by AFF_1:6;
      LIN b,d,b by AFF_1:7;
      hence thesis by A11;
    end;
    hence thesis by A7;
  end;
  now
    assume a=b;
    then LIN a,c,a & LIN b,d,a by AFF_1:7;
    hence thesis;
  end;
  hence thesis by A3;
end;
